Hybrid viscosity implicit scheme for variational inequalities over the fixed point set of an asymptotically nonexpansive mapping in the intermediate sense in Banach spaces

2021 ◽  
Vol 160 ◽  
pp. 296-312
Author(s):  
Rajat Vaish ◽  
Md. Kalimuddin Ahmad
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Nonexpansive Mapping ◽  
Banach Spaces ◽  
Implicit Scheme ◽  
Asymptotically Nonexpansive Mapping ◽  
Fixed Point Set ◽  
Asymptotically Nonexpansive ◽  
Point Set

scholarly journals Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Zhao-Rong Kong ◽  
Lu-Chuan Ceng ◽  
Qamrul Hasan Ansari ◽  
Chin-Tzong Pang
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Variational Inequality Problem ◽  
Extragradient Method ◽  
Fixed Point Set ◽  
Inequality Problem ◽  
Hybrid Extragradient Method ◽  
Point Set ◽  
Hierarchical Variational Inequalities

We consider a triple hierarchical variational inequality problem (THVIP), that is, a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities (SHVI), that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI.

scholarly journals Strong Convergence Iterative Algorithms for Equilibrium Problems and Fixed Point Problems in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Shenghua Wang ◽  
Shin Min Kang
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Iterative Algorithms ◽  
Common Fixed Points ◽  
Common Element ◽  
Fixed Point Set ◽  
Point Set

We first introduce the concept of Bregman asymptotically quasinonexpansive mappings and prove that the fixed point set of this kind of mappings is closed and convex. Then we construct an iterative scheme to find a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a countable family of Bregman asymptotically quasinonexpansive mappings in reflexive Banach spaces and prove strong convergence theorems. Our results extend the recent ones of some others.

scholarly journals On strong convergence theorems for a viscosity-type extragradient method

Filomat ◽  
2021 ◽  
Vol 35 (3) ◽  
pp. 1033-1043
Author(s):  
L.C. Ceng ◽  
C.S. Fong
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Extragradient Method ◽  
Monotone Mapping ◽  
Fixed Point Problem ◽  
Asymptotically Nonexpansive Mapping ◽  
Inclusion Problem ◽  
Point Problem ◽  
Asymptotically Nonexpansive

In this paper, we introduce a general viscosity-type extragradient method for solving the fixed point problem of an asymptotically nonexpansive mapping and the variational inclusion problem with two accretive operators. We obtain a strong convergence theorem in the setting of Banach spaces. In terms of this theorem, we establish the strong convergence result for solving the fixed point problem (FPP) of an asymptotically nonexpansive mapping and the variational inequality problem (VIP) for an inverse-strongly monotone mapping in the framework of Hilbert spaces. Finally, this result is applied to deal with the VIP and FPP in an illustrating example.

scholarly journals STRONG CONVERGENCE OF A HYBRID ITERATION FOR A GENERALIZED MIXED EQUILIBRIUM PROBLEM AND A BREGMAN TOTALLY QUASI-ASYMPTOTICALLY NONEXPANSIVE MAPPING IN BANACH SPACES

2021 ◽  
Vol 18 (9) ◽  
pp. 1620
Author(s):  
Nguyễn Trung Hiếu
Keyword(s):  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Equilibrium Problem ◽  
Banach Spaces ◽  
Asymptotically Nonexpansive Mapping ◽  
Generalized Mixed Equilibrium Problem ◽  
Mixed Equilibrium Problem ◽  
Asymptotically Nonexpansive ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

 Mục đích của nghiên cứu này là kết hợp khoảng cách Bregman với phương pháp chiếu thu hẹp để giới thiệu một dãy lặp lai ghép mới cho bài toán cân bằng hỗn hợp tổng quát và ánh xạ tựa tiệm cận không giãn hoàn toàn Bregman. Sau đó, với những điều kiện thích hợp, chúng tôi chứng minh rằng dãy lặp được đề xuất hội tụ mạnh đến hình chiếu Bregman của điểm xuất phát lên giao của tập nghiệm bài toán cân bằng hỗn hợp tổng quát và tập điểm bất động của ánh xạ tựa tiệm cận không giãn hoàn toàn Bregman trong không gian Banach phản xạ. Định lí này cải tiến kết quả trong (Alizadeh & Moradlou, 2016) từ ánh xạ lai ghép tổng quát và bài toán cân bằng trong không gian Hilbert sang ánh xạ tựa tiệm cận không giãn hoàn toàn Bregman và bài toán cân bằng hỗn hợp tổng quát trong không gian Banach phản xạ. Kết quả được áp dụng cho bài toán cân bằng hỗn hợp tổng quát và ánh xạ tựa tiệm cận không giãn Bregman trong không gian Banach phản xạ. Đồng thời, một ví dụ được đưa ra để minh họa cho dãy lặp được đề xuất. 

Examples and Questions in the Theory of Fixed Point Sets

1979 ◽  
Vol 31 (5) ◽  
pp. 1017-1032 ◽  
Author(s):  
John R. Martin ◽  
Sam B. Nadler
Keyword(s):  
Fixed Point ◽  
Continuous Function ◽  
Banach Spaces ◽  
Metric Spaces ◽  
Compact Manifolds ◽  
Invariance Property ◽  
Point Sets ◽  
Fixed Point Set ◽  
Point Set ◽  
Fixed Point Sets

All spaces considered in this paper will be metric spaces. A subset A of a space X is called a fixed point set of X if there is a map (i.e., continuous function) ƒ: X → X such that ƒ(x) = x if and only if x ∈ A. In [22] L. E. Ward, Jr. defines a space X to have the complete invariance property (CIP) provided that each of the nonempty closed subsets of X is a fixed point set of X. The problem of determining fixed point sets of spaces has been investigated in [14] through [20] and [22]. Some spaces known to have CIP are n-cells[15], dendrites [20], convex subsets of Banach spaces [22], compact manifolds without boundary [16], and a class of polyhedra which includes all compact triangulable manifolds with or without boundary [18].

scholarly journals Strong Convergence of an Iterative Algorithm for Hierarchical Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Poom Kumam ◽  
Thanyarat Jitpeera
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Iterative Algorithm ◽  
Variational Inequality Problem ◽  
Fixed Point Set ◽  
Mild Conditions ◽  
Hierarchical Problems ◽  
Point Set

We introduce the triple hierarchical problem over the solution set of the variational inequality problem and the fixed point set of a nonexpansive mapping. The strong convergence of the algorithm is proved under some mild conditions. Our results extend those of Yao et al., Iiduka, Ceng et al., and other authors.

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